FRG: Collaborative Research: Birational Geometry and Singularities in Zero and Positive Characteristic

这个项目的目标是聚集一群在高维代数几何的广泛主题方面具有经验的研究人员，以便在两个密切相关的领域取得进展：正特征的二次几何以及最小模型程序中出现的奇点和线性级数理论。虽然我们对高维簇几何的理解最近取得了许多进展，但几乎所有这些工作都被限制在特征零上，这是因为使用了可能在正特征上失败的消失定理。这项合作研究的第一个目标是建立一个工具和框架，使双曲面几何的主要结果能够扩展到积极的特征。这将系统地利用最近设计的技术来利用弗罗贝尼乌斯态射。该项目的第二个目标是进一步发展奇点不变量和线性级数的研究，着眼于最小模型程序中的剩余问题。最近在这一领域取得了许多进展。特别地，Shokurov断言某些奇点的不变量(对数正则阈值)满足所谓的ACC性质的猜想已经被一些PI解决。这表明，其他相关但更难的问题可能是触手可及的，这些问题的重要性来自于与最小模型程序中剩余的猜想之一--翻转的终止--的联系。PI建议解决其中一个问题，预测另一个不变量的ACC性质，即最小对数差异。在另一个方向上，PI计划从各种正不变量的观点出发，对代数簇上表现出病理行为的线性系统的例子进行系统的研究。在过去的十年中，高维代数簇的研究取得了重大突破，但仍有几个重要问题有待解决。这样一个中心问题与奇点的研究密切相关，这个项目的目标之一是在理解出现在这一背景下的奇点不变量的性质方面取得进展。PIS的另一个总体目标是系统地发展至少3维正特征的代数簇的研究。在这种新现象(有时被认为是病态的)出现的背景下，人们所知的要少得多。PI预计，思想与其他领域的交叉授粉，特别是与交换代数的交叉授粉，将在这一方向取得进展方面发挥重要作用。此外，在这种情况下的技术和结果很可能在其他领域(例如，在算术几何中)有许多应用。作为合作努力的一部分，私人投资机构计划举办几项活动，将致力于相关问题的数学界成员聚集在一起，并帮助传播作为该项目一部分开发的结果和技术。

The goal of this project is to bring together a group of researchers with experience in a broad array of topics in higher dimensional algebraic geometry, in order to make progress in two closely related areas: birational geometry in positive characteristics and the theory of singularities and linear series arising in the minimal model program. While there has been a lot of recent progress in our understanding of the geometry of higher-dimensional varieties, almost all of this body of work is restricted to characteristic zero, due to the use of vanishing theorems that can fail in positive characteristic. The first goal of this collaborative research is to build tools and a framework that would allow the main results in birational geometry to be extended to positive characteristics. This would make systematic use of the recent techniques that have been devised to exploit the Frobenius morphism. A second goal of the project is to further develop the study of invariants of singularities and of linear series, with an eye toward the remaining problems in the minimal model program. There have been many recent advances in this area. In particular, a conjecture of Shokurov asserting that certain invariants of singularities (the log canonical thresholds) satisfy the so-called ACC property has been solved by some of the PIs. This suggests that other related but harder questions might be within reach, questions whose importance comes from the connection with one of the remaining conjectures in the minimal model program, the termination of flips. The PIs propose to attack one of these problems, predicting the ACC property of another invariant, the minimal log discrepancy. In a separate direction, the PIs plan to undertake a systematic study of examples of linear systems on algebraic varieties that exhibit a pathological behavior from the point of view of various positivity invariants.The last ten years have seen major breakthroughs in the study of higher-dimensional algebraic varieties, but several important problems are still open. A central such problem is intimately related to the study of singularities and one of the goals of this project is to make progress on understanding the properties of the invariants of singularities that appear in this setting. Another general goal of the PIs is to develop systematically the study of algebraic varieties of dimension at least 3 in positive characteristic. A lot less is known in this setting, where new phenomena (sometimes considered pathological) arise. The PIs expect that cross-pollination of ideas with other areas, in particular with commutative algebra, will play an important role in making progress in this direction. Moreover, it is likely that techniques and results in this context would have many applications to other fields (for example, in arithmetic geometry). As part of the collaborative effort, the PIs plan several events that will bring together members of the mathematical community working on related problems and also help disseminate the results and the techniques developed as part of this project.